Jonas Krampe

Jonas Krampe, Efstathios Paparoditis, Carsten Trenkler

We consider statistical inference for impulse responses in sparse, structural high-dimensional vector autoregressive (SVAR) systems. We introduce consistent estimators of impulse responses in the high-dimensional setting and suggest valid inference procedures for the same parameters. Statistical inference in our setting is much more involved since standard procedures, like the delta-method, do not apply. By using local projection equations, we first construct a de-sparsified version of regularized estimators of the moving average parameters associated with the VAR system. We then obtain estimators of the structural impulse responses by combining the aforementioned de-sparsified estimators with a non-regularized estimator of the contemporaneous impact matrix, also taking into account the high-dimensionality of the system. We show that the distribution of the derived estimators of structural impulse responses has a Gaussian limit. We also present a valid bootstrap procedure to estimate this distribution. Applications of the inference procedure in the construction of confidence intervals for impulse responses as well as in tests for forecast error variance decomposition are presented. Our procedure is illustrated by means of simulations.

Jonas Krampe, Efstathios Paparoditis

High-dimensional vector autoregressive (VAR) models are important tools for the analysis of multivariate time series. This paper focuses on high-dimensional time series and on the different regularized estimation procedures proposed for fitting sparse VAR models to such time series. Attention is paid to the different sparsity assumptions imposed on the VAR parameters and how these sparsity assumptions are related to the particular consistency properties of the estimators established. A sparsity scheme for high-dimensional VAR models is proposed which is found to be more appropriate for the time series setting considered. Furthermore, it is shown that, under this sparsity setting, threholding extents the consistency properties of regularized estimators to a wide range of matrix norms. Among other things, this enables application of the VAR parameters estimators to different inference problems, like forecasting or estimating the second-order characteristics of the underlying VAR process. Extensive simulations compare the finite sample behavior of the different regularized estimators proposed using a variety of performance criteria.